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ATOMIC BONDING,ARRANGEMENT ANDSTRUCTURE

Dr. Deni Ferdian, M.Sc

• Quantum numbers are the numbers that assign electrons in anatom to discrete energy levels.

• A quantum shell is a set of fixed energy levels to which electronsbelong.

• Pauli exclusion principle specifies that no more than two electronsin a material can have the same energy. The two electrons haveopposite magnetic spins.

• The valence of an atom is the number of electrons in an atom thatparticipate in bonding or chemical reactions.

• Electronegativity describes the tendency of an atom to gain anelectron.

The Electronic Structure of the Atom

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Nucleus: Z = # protons

orbital electrons:n = principalquantum number

n=3 2 1

= 1 for hydrogen to 94 for plutoniumN = # neutrons

Atomic mass A ≈ Z + N

Bohr – Model (1922)

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The atomic structure of sodium, atomic number 11, showing the electronsin the K, L, and M quantum shells

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son Learning™

The complete set of quantum numbers for each of the 11 electrons insodium

• have complete s and p subshells• tend to be unreactive.

Stable electron configurations...

Z Element Configuration

2 He 1s2

10 Ne 1s22s22p6

18 Ar 1s22s22p63s23p6

36 Kr 1s22s22p63s23p63d104s24p6

Stable Electron Configurations

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• Why? Valence (outer) shell usually not filled completely.

• Most elements: Electron configuration not stable.ElementHydrogenHeliumLithiumBerylliumBoronCarbon...NeonSodiumMagnesiumAluminum...Argon...Krypton

Atomic #123456

10111213

18...36

Electron configuration1s1

1s2 (stable)1s22s1

1s22s2

1s22s22p1

1s22s22p2

...1s22s22p6 (stable)1s22s22p63s1

1s22s22p63s2

1s22s22p63s23p1

...1s22s22p63s23p6 (stable)...1s22s22p63s23p63d104s246 (stable)

Adapted from Table 2.2,Callister 6e.

• III-V semiconductor is a semiconductor that is based on group 3A and5B elements (e.g. GaAs).

• II-VI semiconductor is a semiconductor that is based on group 2B and6B elements (e.g. CdSe).

• Transition elements are the elements whose electronic configurationsare such that their inner “d” and “f” levels begin to fill up.

• Electropositive element is an element whose atoms want toparticipate in chemical interactions by donating electrons and aretherefore highly reactive.

The Periodic Table

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Figure Periodic Table of Elements

• Columns: Similar Valence Structure

Electropositive elements:Readily give up electronsto become + ions.

Electronegative elements:Readily acquire electronsto become - ions.

He

Ne

Ar

Kr

Xe

Rn

ine

rt g

ase

sa

cc

ep

t 1

ea

cc

ep

t 2

e

giv

e u

p 1

eg

ive

up

2e

giv

e u

p 3

e

FLi Be

Metal

Nonmetal

Intermediate

H

Na Cl

Br

I

At

O

SMg

Ca

Sr

Ba

Ra

K

Rb

Cs

Fr

Sc

Y

Se

Te

Po

Adaptedfrom Fig. 2.6,Callister 6e.

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• Ranges from 0.7 to 4.0,

Smaller electronegativity Larger electronegativity

He-

Ne-

Ar-

Kr-

Xe-

Rn-

F4.0

Cl3.0

Br2.8

I2.5

At2.2

Li1.0

Na0.9

K0.8

Rb0.8

Cs0.7

Fr0.7

H2.1

Be1.5

Mg1.2

Ca1.0

Sr1.0

Ba0.9

Ra0.9

Ti1.5

Cr1.6

Fe1.8

Ni1.8

Zn1.8

As2.0

• Large values: tendency to acquire electrons.

Adapted from Fig. 2.7, Callister 6e. (Fig. 2.7 is adapted from Linus Pauling, The Nature of theChemical Bond, 3rd edition, Copyright 1939 and 1940, 3rd edition. Copyright 1960 by CornellUniversity.

Electronegativity©

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The electronegativities of selected elements relative to the position ofthe elements in the periodic table

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Atomic Bonding

Na (metal)unstable

Cl (nonmetal)unstable

electron

+ -CoulombicAttraction

Na (cation)stable

Cl (anion)stable

• Occurs between + and - ions.

• Requires electron transfer.

• Large difference in electronegativity required.

• Example: NaCl

Ionic Bonding

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Ionic bond – metal + nonmetal

donates acceptselectrons electrons

Dissimilar electronegativities

ex: MgO Mg 1s2 2s2 2p6 3s2 O 1s2 2s2 2p4

[Ne] 3s2

Mg2+ 1s2 2s2 2p6 O2- 1s2 2s2 2p6

[Ne] [Ne]

• Predominant bonding in Ceramics

Give up electrons Acquire electrons

He-

Ne-

Ar-

Kr-

Xe-

Rn-

F4.0

Cl3.0

Br2.8

I2.5

At2.2

Li1.0

Na0.9

K0.8

Rb0.8

Cs0.7

Fr0.7

H2.1

Be1.5

Mg1.2

Ca1.0

Sr1.0

Ba0.9

Ra0.9

Ti1.5

Cr1.6

Fe1.8

Ni1.8

Zn1.8

As2.0

CsCl

MgO

CaF2

NaCl

O3.5

Adapted from Fig. 2.7, Callister 6e. (Fig. 2.7 is adapted from Linus Pauling, The Nature of theChemical Bond, 3rd edition, Copyright 1939 and 1940, 3rd edition. Copyright 1960 by CornellUniversity.

Examples: Ionic Bonding

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• Coordination # increases withIssue: How many anions can you

arrange around a cation?

rcationranion

rcationranion

Coord #

< .155

.155-.225

.225-.414

.414-.732

.732-1.0

ZnS(zincblende)

NaCl(sodium

chloride)

CsCl(cesium

chloride)

2

3

4

6

8Adapted from Table12.2, Callister 6e.

Adapted from Fig. 12.2,Callister 6e.

Adapted from Fig. 12.3,Callister 6e.

Adapted from Fig. 12.4,Callister 6e.

Coordination # And Ionic Radii

• Requires shared electrons

• Example: CH4

C: has 4 valence e,needs 4 more

H: has 1 valence e,needs 1 more

Electronegativitiesare comparable.

shared electronsfrom carbon atom

shared electronsfrom hydrogenatoms

H

H

H

H

C

CH4

Adapted from Fig. 2.10, Callister 6e.

Covalent Bonding

Covalent bonding requires that electrons be shared between atoms insuch a way that each atom has its outer sp orbital filled. In silicon, witha valence of four, four covalent bonds must be formed

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• Molecules with nonmetals• Molecules with metals and nonmetals• Elemental solids (RHS of Periodic Table)• Compound solids (about column IVA)

He-

Ne-

Ar-

Kr-

Xe-

Rn-

F4.0

Cl3.0

Br2.8

I2.5

At2.2

Li1.0

Na0.9

K0.8

Rb0.8

Cs0.7

Fr0.7

H2.1

Be1.5

Mg1.2

Ca1.0

Sr1.0

Ba0.9

Ra0.9

Ti1.5

Cr1.6

Fe1.8

Ni1.8

Zn1.8

As2.0

SiC

C(diamond)

H2O

C2.5

H2

Cl2

F2

Si1.8

Ga1.6

GaAs

Ge1.8

O2.0

co

lum

n IV

A

Sn1.8Pb1.8

Adapted from Fig. 2.7, Callister 6e. (Fig. 2.7 isadapted from Linus Pauling, The Nature of the Chemical Bond, 3rd edition, Copyright 1939 and 1940, 3rd edition.Copyright 1960 by Cornell University.

Examples: Covalent Bonding

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Covalent bonds are directional. In silicon, a tetrahedral structure isformed, with angles of 109.5° required between each covalent bond

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© 2003 Brooks/Cole Publishing / ThomsonLearning™

(a) In polyvinyl chloride(PVC), the chlorine atomsattached to the polymerchain have a negative chargeand the hydrogen atoms arepositively charged. Thechains are weakly bonded byvan der Waals bonds. Thisadditional bonding makesPVC stiffer, (b) When a forceis applied to the polymer, thevan der Waals bonds arebroken and the chains slidepast one another

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The metallic bond formswhen atoms give uptheir valence electrons,which then form anelectron sea. Thepositively charged atomcores are bonded bymutual attraction to thenegatively chargedelectrons

Metallic Bonding

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rooks/Cole Publishing / Thom

son Learning™

Figure When voltage is applied to a metal, the electrons in the electronsea can easily move and carry a current

Arises from interaction between dipoles

• Permanent dipoles-molecule induced

• Fluctuating dipoles

+ - secondarybonding + -

H Cl H Clsecondarybonding

secondary bonding

HH HH

H2 H2

secondarybonding

ex: liquid H2asymmetric electronclouds

+ - + -secondary

bonding

-general case:

-ex: liquid HCl

-ex: polymer

Adapted from Fig. 2.13, Callister 6e.

Adapted from Fig. 2.14,Callister 6e.

Adapted from Fig. 2.14,Callister 6e.

Secondary Bonding

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TypeIonic

Covalent

Metallic

Secondary

Bond EnergyLarge!

Variable

large-Diamond

small-Bismuth

Variable

large-Tungsten

small-Mercury

smallest

CommentsNondirectional (ceramics)

Directional

semiconductors, ceramics

polymer chains)

Nondirectional (metals)

Directionalinter-chain (polymer)

inter-molecular

Bonding Type

• Interatomic spacing is the equilibrium spacing between the centers oftwo atoms.

• Binding energy is the energy required to separate two atoms fromtheir equilibrium spacing to an infinite distance apart.

• Modulus of elasticity is the slope of the stress-strain curve in theelastic region (E).

• Yield strength is the level of stress above which a material begins toshow permanent deformation.

• Coefficient of thermal expansion (CTE) is the amount by which amaterial changes its dimensions when the temperature changes.

Binding Energy and Interatomic Spacing

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Atoms or ions areseparated by andequilibrium spacing thatcorresponds to theminimum inter-atomicenergy for a pair of atomsor ions (or when zeroforce is acting to repel orattract the atoms or ions)

• Bond length, r

• Bond energy, Eo

FF

r

• Melting Temperature, Tm

Eo=

“bond energy”

Energy (r)

ro runstretched length

r

larger Tm

smaller Tm

Energy (r)

ro

Tm is larger if Eo is larger.

PROPERTIES FROM BONDING: TM

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• Elastic modulus, E

• E ~ curvature at ro

crosssectionalarea Ao

L

length, Lo

F

undeformed

deformed

LFAo

= ELo

Elastic modulus

r

larger Elastic Modulus

smaller Elastic Modulus

Energy

rounstretched length

E is larger if Eo is larger.

Properties From Bonding: E

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The force-distance curve for two materials, showing the relationshipbetween atomic bonding and the modulus of elasticity, a steep dFldaslope gives a high modulus

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• Coefficient of thermal expansion, α

• α ~ symmetry at ro

α is larger if Eo is smaller.

L

length, Lo

unheated, T1

heated, T2= (T2-T1)L

Lo

coeff. thermal expansion

r

smaller

larger

Energy

ro

Properties From Bonding: α

The inter-atomic energy (IAE)-separation curve for two atoms. Materialsthat display a steep curve with a deep trough have low linear coefficientsof thermal expansion

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Ceramics

(Ionic & covalent bonding):

Metals

(Metallic bonding):

Polymers

(Covalent & Secondary):

secondary bonding

Large bond energylarge Tm

large Esmall α

Variable bond energymoderate Tm

moderate Emoderate α

Directional PropertiesSecondary bonding dominates

small Tsmall Elarge α

Summary: Primary Bonds

Level of Structure Example of Technologies

Atomic Structure Diamond – edge ofcutting tools

Atomic Arrangements: Lead-zirconium-titanateLong-Range Order [Pb(Zrx Ti1-x )] or PZT –(LRO) gas igniters

Atomic Arrangements: Amorphous silica - fiberShort-Range Order optical communications(SRO) industry

Levels of Structure

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Level of Structure Example of Technologies

Nanostructure Nano-sized particles ofiron oxide – ferrofluids

Microstructure Mechanical strength ofmetals and alloys

Macrostructure Paints for automobilesfor corrosion resistance

(Continued)

CRYSTAL STRUCTURE OF SOLIDS

NaCl

Quartz Crystal

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• Non dense, random packing

• Dense, ordered packing

Dense, ordered packed structures tend to have lower energies.

Energy and PackingEnergy

r

typical neighborbond length

typical neighborbond energy

Energy

r

typical neighborbond length

typical neighborbond energy

CO

OLI

NG

• atoms pack in periodic, 3D arrays• typical of:

Crystalline materials...

-metals-many ceramics-some polymers

• atoms have no periodic packing• occurs for:

Noncrystalline materials...

-complex structures-rapid cooling

Si Oxygen

crystalline SiO2

noncrystalline SiO2"Amorphous" = NoncrystallineAdapted from Fig. 3.18(b),Callister 6e.

Adapted from Fig. 3.18(a),Callister 6e.

Materials and Packing

LONG RANGE ORDER

SHORT RANGE ORDER

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Metallic Crystal Structures• How can we stack metal atoms to minimize empty

space?2-dimensions

vs.

Now stack these 2-D layers to make 3-D structures

Robert Hooke – 1660 - Cannonballs“Crystal must owe its regular shape to the

packing of spherical particles”

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Niels Steensen ~ 1670observed that quartz crystals hadthe same angles betweencorresponding faces regardless oftheir size.

Christian Huygens - 1690

42

Studying calcite crystalsmade drawings of atomicpacking and bulk shape.

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Early Crystallography

René-Just Haüy (1781): cleavage of calcite• Common shape to all shards: rhombohedral• How to model this mathematically?• What is the maximum number of distinguishable shapes

that will fill three space?• Mathematically proved that there are only 7 distinct

space-filling volume elements

Lattice - A collection of points that divide space into smallerequally sized segments.

Basis - A group of atoms associated with a lattice point. Unit cell - A subdivision of the lattice that still retains the overall

characteristics of the entire lattice. Atomic radius - The apparent radius of an atom, typically

calculated from the dimensions of the unit cell, using close-packed directions (depends upon coordination number).

Packing factor - The fraction of space in a unit cell occupied byatoms.

Lattice, Unit Cells, Basis, and CrystalStructures

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Prof. Bondan T. Sofyan

There are two kinds of crystalline solid:• Single crystal, where ALL atoms in that material arrange

themselves in one direction only.• Polycrystal. This material consists of several group of

atoms (grains) that have different orientation to eachother.

Lattice

Prof. Bondan T. Sofyan

• As explained before, a three-dimensional periodicarrangement of atoms, ions or molecules is alwayspresent in all crystals. If each atom is represented by apoint (its centre of gravity), the arrangement is called alattice.

Three-dimensional periodicarrangement of atoms in a crystal

The lattice of the crystal

A Lattice is a three dimensionalarrangement of points in which all of thepoints have identical surroundings

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Crystallographic Points, Directions, and Planes

• It is necessary to specify a particularpoint/location/atom/direction/plane in a unit cell

• We need some labeling convention. Simplest way is touse a 3-D system, where every location can beexpressed using three numbers or indices.• a, b, c and α, β, γ

x

y

z

βα

γ

Point Coordinates – Atom PositionsPoint coordinates for unit cell

center area/2, b/2, c/2 ½ ½½

Point coordinates for unit cellcorner are 111

Translation: integer multiple oflattice constants identicalposition in another unit cell

z

x

ya b

c

000

111

y

z

2c

b

b

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Crystallographic Directions

1. Vector repositioned (if necessary) to passthrough origin.

2. Read off projections in terms ofunit cell dimensions a, b, and c

3. Adjust to smallest integer values4. Enclose in square brackets, no commas

[uvw]

ex: 1, 0, ½ => 2, 0, 1 => [ 201 ]

-1, 1, 1

families of directions <uvw>

z

x

Algorithm

where overbar represents anegative index

[ 111 ]=>

y

ex: linear density of Al in [110]direction

a = 0.405 nm

Linear Density• Linear Density of Atoms LD =

a

[110]

Unit length of direction vectorNumber of atoms

# atoms

length

13.5 nma2

2LD −==

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Crystallographic Planes

Adapted from Fig. 3.9, Callister 7e.

Crystallographic Planes• Miller Indices: Reciprocals of the (three) axialintercepts for a plane, cleared of fractions &common multiples. All parallel planes have sameMiller indices.

• Algorithm1. If plane passes thru origin, translate2. Read off intercepts of plane with axes in terms of a, b,

c3. Take reciprocals of intercepts4. Reduce to smallest integer values5. Enclose in parentheses, no commas i.e., (hkl)

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Crystallographic Planesz

x

ya b

c

4. Miller Indices (110)

example a b cz

x

ya b

c

4. Miller Indices (100)

1. Intercepts 1 1 2. Reciprocals 1/1 1/1 1/

1 1 03. Reduction 1 1 0

1. Intercepts 1/2 2. Reciprocals 1/½ 1/ 1/

2 0 03. Reduction 2 0 0

example a b c

Crystallographic Planesz

x

ya b

c

4. Miller Indices (634)

example1. Intercepts 1/2 1 3/4

a b c

2. Reciprocals 1/½ 1/1 1/¾

2 1 4/3

3. Reduction 6 3 4

(001)(010),

Family of Planes {hkl}

(100), (010),(001),Ex: {100} = (100),

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Crystallographic Planes• We want to examine the atomic packing of

crystallographic planes• Iron foil can be used as a catalyst. The

atomic packing of the exposed planes isimportant.

a) Draw (100) and (111) crystallographic planesfor Fe.

b) Calculate the planar density for each of theseplanes.

The Seven Crystal Systems

Specification of unit cellparameters

BASIC UNIT

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August Bravais• How many different ways can I put atoms into these seven

crystal systems, and get distinguishable point environments?

And, he proved mathematically that there are 14 distinct ways to arrange points inspace.

When I start puttingatoms in the cube, I havethree distinguishablearrangements.

SC BCC FCC

Crystal Systems

7 crystal systems

14 crystal lattices

Fig. 3.4, Callister 7e.

Unit cell: smallest repetitive volume whichcontains the complete lattice pattern of a crystal.

a, b, and c are the lattice constants

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The fourteen types ofBravais latticesgrouped in sevencrystal systems.

Cubic Three equal axes at rightangles.a=b=c. ===90

SimpleBody-centredFace-centred

PIF

Tetragonal Three equal axes at rightangles, two equal.a=bc. ===90

SimpleBody-centred

PI

Orthorhombic Three unequal axes at rightangles.abc. ===90

SimpleBody-centredBase-centredFace-centred

PICF

Rhombohedral(trigonal)

3 equal axes, equally inclined.a=b=c. ==90

Simple R

Hexagonal Two equal, coplanar axes at120, 3rd axis at right angles.a=bc. ==90, =120

Simple P

Monoclinic Three unequal axes, one pairnot at right angles.abc. ==90

SimpleBase-centred

PC

Triclinic Three unequal axes, unequallyinclined, none at right angles.abc. 90

Simple P

Unit Cell

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• Rare due to poor packing• Close-packed directions are cube edges.

• Coordination # = 6(# nearest neighbors)

Simple Cubic Structure (SC)

Closed packed direction is wherethe atoms touch each other

• Coordination # = 8

(Courtesy P.M. Anderson)

• Close packed directions are cube diagonals.

--Note: All atoms are identical; the center atom is shadeddifferently only for ease of viewing.

Body Centered Cubic Structure (BCC)

ex: Cr, W, Fe (), Tantalum, Molybdenum

2 atoms/unit cell: 1 center + 8 corners x 1/8

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APF =Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

• APF for a simple cubic structure = 0.52

APF =a3

4

3(0.5a)31

atoms

unit cellatom

volume

unit cellvolume

close-packed directions

a

R=0.5a

contains 8 x 1/8 =1 atom/unit cell

Atomic Packing Factor : BCC

(Courtesy P.M. Anderson)

• Close packed directions are face diagonals.

--Note: All atoms are identical; the face-centered atoms are shadeddifferently only for ease of viewing.

Face Centered Cubic Structure (FCC)

• Coordination # = 12

Adapted from Fig. 3.1, Callister 7e.

ex: Al, Cu, Au, Pb, Ni, Pt, Ag

4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8

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APF =a3

4

3( 2a/4)34

atoms

unit cell atomvolume

unit cell

volume

Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell

a

• APF for a body-centered cubic structure = 0.74

Close-packed directions:length = 4R

= 2 a

Atomic Packing Factor: FCC

Characteristics of Cubic Lattices

Unit Cell Volume a3 a3 a3

Lattice Points per cell 1 2 4

Nearest Neighbor Distance a a√3/2 a√2/2Number of Nearest Neighbors 6 8 12

Atomic Packing Factor 0.52 0.68 0.74

66

BCC FCCSC

APF =Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

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Example: Copper

n AVcNA

# atoms/unit cell Atomic weight (g/mol)

Volume/unit cell

(cm3/unit cell)Avogadro's number

(6.023 x 1023 atoms/mol)

• crystal structure = FCC: 4 atoms/unit cell• atomic weight = 63.55 g/mol (1 amu = 1 g/mol)• atomic radius R = 0.128 nm (1 nm = 10 cm)

-7

Vc = a3 ; For FCC, a = 4R/ 2 ; Vc = 4.75 x 10-23cm3

Compare to actual: Cu = 8.94 g/cm3Result: theoretical Cu = 8.89 g/cm3

Theoretical Density, ρ

ElementAluminumArgonBariumBerylliumBoronBromineCadmiumCalciumCarbonCesiumChlorineChromiumCobaltCopperFlourineGalliumGermaniumGoldHeliumHydrogen

SymbolAlArBaBeBBrCdCaCCsClCrCoCuFGaGeAuHeH

At. Weight(amu)26.9839.95137.339.01210.8179.90112.4140.0812.011132.9135.4552.0058.9363.5519.0069.7272.59196.974.0031.008

Atomic radius(nm)0.143------0.2170.114------------0.1490.1970.0710.265------0.1250.1250.128------0.1220.1220.144------------

Density(g/cm3)2.71------3.51.852.34------8.651.552.251.87------7.198.98.94------5.905.3219.32------------

CrystalStructureFCC------BCCHCPRhomb------HCPFCCHexBCC------BCCHCPFCC------Ortho.Dia. cubicFCC------------

Characteristics of Selected Elements at 20 deg C

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metals ceramics polymers

(g

/cm

3)

Graphite/Ceramics/Semicond

Metals/Alloys

Composites/fibersPolymers

1

2

20

30Based on data in Table B1, Callister

*GFRE, CFRE, & AFRE are Glass,Carbon, & Aramid Fiber-ReinforcedEpoxy composites (values based on

60% volume fraction of aligned fibersin an epoxy matrix).10

345

0.30.40.5

Magnesium

Aluminum

Steels

Titanium

Cu,Ni

Tin, Zinc

Silver, Mo

TantalumGold, WPlatinum

GraphiteSilicon

Glass-sodaConcrete

Si nitrideDiamondAl oxide

Zirconia

HDPE, PSPP, LDPE

PC

PTFE

PETPVCSilicone

Wood

AFRE*

CFRE*

GFRE*

Glass fibers

Carbon fibers

Aramid fibers

Why?Metals have...• close-packing

(metallic bonding)• large atomic mass

Ceramics have...• less dense packing

(covalent bonding)• often lighter elements

Polymers have...• poor packing

(often amorphous)• lighter elements (C,H,O)

Composites have...• intermediate values

Data from Table B1, Callister 6e.

Densities Of Material Classes

Polymorphism & Allotropy• Some materials may exist in more than one crystal

structure, this is called polymorphism.• If the material is an elemental solid, it is called

allotropy. An example of allotropy is carbon, whichcan exist as diamond, graphite, and amorphouscarbon.

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“The same atoms can have more than one crystalstructure”.

Temperature, C

BCC Stable

FCC Stable

914

1391

1536

shorter

longer!shorter!

longer

Tc 768 magnet falls off

BCC Stable

Liquid

heat up

cool down

Example. Iron (BCC-FCC-BCC)

(c) 2003 Brooks/Cole Publishing / ThomsonLearning™

Figure 3.10 Atomic arrangements in crystalline silicon and amorphoussilicon. (a) Amorphous silicon. (b) Crystalline silicon. Note the variation in theinter-atomic distance for amorphous silicon.

Amorphous Materials

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• Single Crystals

-Properties vary withdirection: anisotropic.

-Example: the modulusof elasticity (E) in BCC iron:

• Polycrystals

-Properties may/may notvary with direction.

-If grains are randomlyoriented: isotropic.(Epoly iron = 210 GPa)

-If grains are textured,anisotropic.

E (diagonal) = 273 GPa

E (edge) = 125 GPa

200 µm

Data from Table 3.3,Callister 6e.(Source of data isR.W. Hertzberg,Deformation andFracture Mechanics ofEngineering Materials,3rd ed., John Wileyand Sons, 1989.)

Adapted from Fig.4.12(b), Callister 6e.(Fig. 4.12(b) iscourtesy of L.C. Smithand C. Brady, theNational Bureau ofStandards,Washington, DC [nowthe National Instituteof Standards andTechnology,Gaithersburg, MD].)

Single vs Polycrystals

Single crystal

Polycrystal

GRAIN

In onegrain,atoms areoriented atthe samedirection

Prof. Bondan T. Sofyan

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• Most engineering materials are polycrystals.

• Nb-Hf-W plate with an electron beam weld.• Each "grain" is a single crystal.• If crystals are randomly oriented,

overall component properties are not directional.• Crystal sizes typ. range from 1 nm to 2 cm

(i.e., from a few to millions of atomic layers).

Adapted from Fig. K,color inset pages ofCallister 6e.(Fig. K is courtesy ofPaul E. Danielson,Teledyne Wah ChangAlbany)

1 mm

Polycrystals

Diffraction - The constructive interference, or reinforcement, of abeam of x-rays or electrons interacting with a material. Thediffracted beam provides useful information concerning thestructure of the material.

Bragg’s law - The relationship describing the angle at which abeam of x-rays of a particular wavelength diffracts fromcrystallographic planes of a given interplanar spacing.

In a diffractometer a moving x-ray detector records the 2y anglesat which the beam is diffracted, giving a characteristic diffractionpattern

Diffraction Techniques for CrystalStructure Analysis

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d=n/2sinc

x-rayintensity(fromdetector)

c

• Incoming X-rays diffract from crystal planes.

• Measurement of:Critical angles, θc,for X-rays provideatomic spacing, d.

Adapted from Fig.3.2W, Callister 6e.

X-RAYS - CRYSTAL STRUCTURE

reflections mustbe in phase todetect signal

spacingbetweenplanes

d

incoming

X-rays

outgoin

gX-ra

ys

detector

extradistancetravelledby wave “2”

“1”

“2”

“1”

“2”

Bragg’s law.

Photograph of a XRD diffractometer.(Courtesy of H&M Analytical Services.)

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(c) 2003 Brooks/C

ole Publishing / Thom

son Learning

(a) Diagram of a diffractometer,showing powder sample, incidentand diffracted beams. (b) Thediffraction pattern obtained from asample of gold powder.